In a recent paper published at Advances in High Energy Physics (AHEP) journal, Yang Zhao et al. derived Maxwell equations on Cantor sets from the local fractional vector calculus. It can be shown that Maxwell equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Using the same approach, elsewhere Yang, Baleanu & Tenreiro Machado derived systems of Navier-Stokes equations on Cantor sets. However, so far there is no derivation of Proca equations on Cantor sets. Therefore, in this paper we present for the first time a derivation of Proca equations and GravitoElectroMagnetic (GEM) Proca-type equations on Cantor sets. Considering that Proca equations may be used to explain electromagnetic effects in superconductor, We suggest that Proca equations on Cantor sets can describe electromagnetic of fractal superconductors; besides GEM Proca-type equations on Cantor sets may be used to explain some gravitoelectromagnetic effects of superconductor for fractal media. It is hoped that this paper may stimulate further investigations and experiments in particular for fractal superconductor. It may be expected to have some impact to fractal cosmology modeling too.

Periodical:

Bulletin of Mathematical Sciences and Applications (Volume 10)

Pages:

48-56

Citation:

V. Christianto and B. Rahul, "A Derivation of Proca Equations on Cantor Sets: A Local Fractional Approach", Bulletin of Mathematical Sciences and Applications, Vol. 10, pp. 48-56, 2014

Online since:

November 2014

Authors:

Keywords:

Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

Mandelbrot, B.B. 1989. Fractal geometry: what is it, and what does it do? Proc. R. Soc. Lond. A. 423, 3-16, URL: http: /users. math. yale. edu/~bbm3/web_pdfs/fractalGeometryWhatIsIt. pdf; [1a] Mandelbrot, B.B. Fractals and the Geometry of Nature. Encyclopedia Britannica, URL: http: /users. math. yale. edu/~bbm3/web_pdfs/encyclopediaBritannica. pdf.

Zhao, Y., Baleanu, D., Cattani, C., Cheng, D-F., & Yang, X-J. 2013. Maxwell's equations on Cantor Sets: A Local Fractional Approach. Advances in High Energy Physics Vol. 2013 Article ID 686371, http: /dx. doi. org/10. 1155/2013/686371, http: /downloads. hindawi. com/journals/ahep/2013/686371. pdf.

Ostoja-Starzewski, Martin. 2012. Electromagnetism in anisotropic fractal media. Z. Angew. Math. Phys. (ZAMP) DOI 10. 1007/s00033-012-0230-z . URL: http: /mechanical. illinois. edu/media/uploads/web_sites/82/files/zamp_2012. 20121002. 06b39d642e 987. 85850489. pdf; [3a] Also in arXiv: 1106. 1491 [math-ph], URL: http: /arxiv. org/pdf/1106. 1491. pdf.

Tajmar, M. 2008. Electrodynamics in Superconductor explained by Proca equations. arXiv: cond-mat/0803. 3080.

de Matos, C.J., &Tajmar, M. 2006. Gravitomagnetic London Moment and the Graviton Mass inside a Superconductor. arXiv: gr-qc/0602591.

Gondran, Michel. 2009. Proca equations derived from first principles. arXiv: 0901. 3300 [quantph], URL: http: /arxiv. org/pdf/0901. 3300. pdf.

Blackledge, Jonathan M. 2007. An Approach to Unification using a Linear Systems Model for the Propagation of Broad-band Signals. ISAST Transaction on Electronics and Signal Processing, Vol. 1, No. 1, 2007. URL: http: /eleceng. dit. ie/papers/100. pdf.

Demir, Suleyman. 2013. Space-time algebra for the generalization of gravitational field equations. Pramana Vol. 80 No. 5 (Indian Academy of Sciences), May 2013, 811-823. URL: http: /www. ias. ac. in/pramana/v80/p.811/fulltext. pdf.

Schwinger, Julian, DeRaad, Jr., Lester L., Milton, Kimball A., & Tsai, Wu-yang. 1998. Classical Electrodynamics. Reading, Massachusetts: Perseus Books. 591 p.

Penrose, Roger. 2004. The Road to Reality: A Complete Guide to the Laws of the Universe. London: Jonathan Cape. 1123 p. URL: http: /staff. washington. edu/freitz/penrose. pdf.

Yang, X-J., Baleanu, D., & Tenreiro Machado, J.A. 2013. Systems of Navier-Stokes on Cantor Sets. Mathematical Problems in Engineering Vol. 2013, Article ID 769724, http: /dx. doi. org/10. 1155/2013/769724, URL: http: /www. hindawi. com/journals/mpe/2013/769724/, or URL: http: /recipp. ipp. pt/bitstream/10400. 22/3455/1/ART_TenreiroMachado_2013_DEE. pdf.

Shpenkov, George P. 2013. Dialectical View of the World: The Wave Model (Selected Lectures). Volume I: Philosophical and Mathematical Background. URL: http: /shpenkov. janmax. com/Vol. 1. Dialectics. pdf.

2006. An Elucidation of the Nature of the Periodic Law, Chapter 7 in The Mathematics of the Periodic Table, Rouvray, D. H. and King, R. B., ed., Nova Science Publishers, NY, pp.119-160.

Shpenkov, George P. & Kreidik, Leonid G. 2005. Schrödinger's error in principle. Galilean Electrodynamics Vol. 16, No. 3, 51-56, (2005); URL: http: /shpenkov. janmax. com/Blunders. pdf.

Kreidik, Leonid G., & Shpenkov, George P. 2002. Important Results of Analyzing Foundations of Quantum Mechanics. Galilean Electrodynamics & QED-EAST, Vol. 13, Special Issues No. 2, 23-30; URL: http: /shpenkov. janmax. com/QM-Analysis. pdf.

Christianto, Victor. 2014. A Review of Schrodinger Equation and Classical Wave Equation. Prespacetime Journal, May 2014. URL: www. prespacetime. com. Also available at: http: /vixra. org/abs/1404. 0020.

Tarasov, Vasily E. 2005. Wave Equation for Fractal Solid String. Mod. Phys. Lett. B, Vol. 19, No. 15, 721-728. URL: http: /theory. sinp. msu. ru/~tarasov/PDF/MPLB2005-1. pdf.

Hu, Ming-Sheng, Agarwal, Ravi P., & Yang, Xiao-Jun. 2012. Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String. Abstract and Applied Analysis Vol. 2012, Article ID 567401, 15p. doi: 10. 1155/2012/567401. URL: http: /downloads. hindawi. com/journals/aaa/2012/567401. pdf.

Christianto, Victor. 2014. A derivation of GravitoElectroMagnetic Proca equations in fractional space. Prespacetime Journal, May 2014. URL: www. prespacetime. com.

Rubino, Leonardo. 2012. The Whole Universe in Three Numbers. http: /vixra. org/abs/1205. 0058.

Rubino, Leonardo. 2011. A Dissertation on the Origins of Quantization in the Universe. http: /vixra. org/abs/1112. 0087.

Rubino, Leonardo. 2014. The Sound of the Universe. http: /vixra. org/abs/1402. 0129.

Zhang, Xin-an. 2013. A wave vibration model basing on classical theory interpret the photoelectric effect, Compton effect, atomic hydrogen spectrum formula, as well as the blackbody radiation, in Xin-an Zhang, Researches in the Nature of Quantum Wave. Xiaotong Publisher, 2014. Available upon request at: 876840956@qq. com.

Baleanu, D., Tenreiro Machado, J.A., Cattani, C., Baleanu, M.C., & Yang, X-J. 2014. Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators. Abstract and Applied Analysis Vol. 2014, Article ID 535048, http: /dx. doi. org/10. 1155/2014/535048; http: /downloads. hindawi. com/journals/aaa/2014/535048. pdf.

Thornhill, C.K. 1996. Real or Imaginary Space-Time? Reality or Relativity? Hadronic Journal Suppl. 11, 3: 209-224. URL: http: /vixra. org/abs/0702. 0044.

Leighton, Luke Kenneth Casson. 2014. Expanded Rishon Model Particles. http: /arXiv. org/abs/1403. 0016.

Tu, L-C., Luo, J., & George T. Gillies. 2005. The mass of the photon. Rep. Prog. Phys. 68: 77- 130. (Institute of Physics Publishing), doi: 10. 1088/0034-4885/68/1/R02.

Christianto, Victor. 2014. An outline of cosmology based on interpretation of The Johannine Prologue. Bull. Soc. Math. Services and Standards (BSOMASS), Sept. 2014. URL: www. ijmsea. com.

Cited By:

[1] D. Baleanu, H. Khan, H. Jafari, R. Khan, "On the Exact Solution of Wave Equations on Cantor Sets", Entropy, Vol. 17, p. 6229, 2015

DOI: https://doi.org/10.3390/e17096229[2] H. Jassim, "The Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operator", Abstract and Applied Analysis, Vol. 2016, p. 1, 2016

DOI: https://doi.org/10.1155/2016/2913539